R^2 is contractible, so any two curves are homotopic. That should give you a good idea of how to construct the homotopy.
I'm stuck and I need help, can anybody help me out with this problem,
Let
f : I → R^2 be given by f (s) = (1 + s + s^2 − sin(πs), 2/1+s^2) and let
g : I → R^2 be given by g(s) = (1 + 2s, 2 − s).
Prove that f ≃ g by explicitly constructing a homotopy.
I know that H(s,0) = f(s) and H(s,1) = g(s)